Mathematical Society of Japan Memoirs

Symmetric functions and the Yangian decomposition of the Fock and basic modules of the affine Lie algebra $\hat{\mathfrak{s}\mathfrak{l}}_{N}$

Denis Uglov

Full-text: Open access


The decomposition of the Fock and basic modules of the affine Lie algebra $\mathfrak{sl}_{N}$ into irreducible submodules of the Yangian $Y(\mathfrak{gl}_N$ is constructed. Each of the irreducible submodules admits the unique up to normalization eigenbasis of the maximal commutative subalgebra of the Yangian. The elements of this eigenbasis are identified with specializations of Macdonald symmetric functions where both parameters of the latter approach an Nth primitive root of unity.

Chapter information

Ivan Cherednik, Peter J. Forrester and Denis Uglov, Quantum Many-Body Problems and Representation Theory (Tokyo: The Mathematical Society of Japan, 1998), 183-241

First available in Project Euclid: 17 January 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Copyright © 1998, The Mathematical Society of Japan


Uglov, Denis. Symmetric functions and the Yangian decomposition of the Fock and basic modules of the affine Lie algebra $\hat{\mathfrak{s}\mathfrak{l}}_{N}$. Quantum Many-Body Problems and Representation Theory, 183--241, The Mathematical Society of Japan, Tokyo, Japan, 1998. doi:10.2969/msjmemoirs/00101C030.

Export citation


  • 1. Al-Salam, W., Allaway, W.R. and Askey, R.: Sieved ultraspherical polynomials, Trans.Amer.Math.Soc. 284 39-55 (1984).
  • 2. Bernard, D., Gaudin, M., Haldane, F.D.M. and Pasquier, V.: Yang-Baxter equation in long-range interacting systems, J. Phys., A26 5219-5236 (1993).
  • 3. Bernard, D.,Pasquier, V. and Serban, D.: Spinons in conformal field theory, Nucl. Phys. B428 612-628 (1994).
  • 4. Bouwknegt, P., Ludwig, A. and Schoutens, K.: Spinon bases, Yangian symmetry and fermionic representations of Virasoro characters in conformal field theory, Phys. Lett. 338B 448-456 (1994).
  • 5. Bouwknegt, P. and Schoutens, K.: The $\widehat{SU(n)_{k}}$ WZW models: Spinon decomposition and Yangian structure, Nucl. Phys. B482 345-372 (1996); Spinon decomposition and Yangian structure of $\hat{\mathfrak{s}\mathfrak{l}_{n}}$ modules, (q-alg/9703021).
  • 6. Cherednik, I.V.: A new interpretation of Gelfand-Zetlin bases, Duke Math. J. 54 563-577 (1987).
  • 7. Cherednik, I.V.: A unification of the Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Inv. Math., 106 411-432 (1991).
  • 8. Cherednik, I.V.: Integration of quantum many-body problems by affine Knizhnik-Zamolodchikov equations, Preprint RIMS-776 (1991); Advances in Math. 106 65-95 (1994).
  • 9. Cherednik, I.V.: Double Affine Hecke algebras and Macdonald's conjectures, Annals Math., 141 191-216 (1995); Non-symmetric Macdonald Polynomials, IMRN 10 483-515 (1995).
  • 10. Drinfeld, V.G.: Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 20 62-64 (1986).
  • 11. Drinfeld, V.G.: A new realization of Yangians and quantized affine algebras, Sov. Math. Dokl. 36 212-216 (1988); "Quantum Groups" in Proceedings of the International Congress of Mathematicians, Amer. Math. Soc., Providence, RI, 798-820 (1987).
  • 12. Dunkl, C.F.: Trans. Amer. Math. Soc. 311 167 (1989).
  • 13. Haldane, F.D.M., Ha, Z.N.C., Talstra, J.C., Bernard, D. and Pasquier, V.: Yangian symmetry of integrable quantum chains with long-range interactions and a new description of states in conformal field theory, Phys. Rev. Lett. 69 2012-2025 (1992).
  • 14. Jimbo, M. and Miwa, T.: Solitons and infinite dimensional Lie algebras, Publ. RIMS Kyoto Univ. 19 943-1001 (1983).
  • 15. Kac, V.G. and Raina, A.: Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, World Scientific, Singapore (1987).
  • 16. Kirillov, A.N., Kuniba, A. and Nakanishi, T.: Skew Young diagram method in spectral decomposition of integrable lattice models, Comm. Math. Phys. 185 441-465 (1997).
  • 17. Kulish, P.P., Reshetikhin, N.Yu. and Sklyanin, E.K.: Yang-Baxter Equation and Representation Theory, Lett.Math.Phys. 5 393-403 (1981).
  • 18. Macdonald, I.G.: Affine Hecke algebra and Orthogonal Polynomials, Séminaire Bourbaki, 47 No. 797, 1-18 (1995).
  • 19. Macdonald, I.G.: Symmetric functions and Hall polynomials, 2-nd ed., Clarendon Press (1995).
  • 20. Molev, A., Nazarov, M. and Olshanskiĭ G.: Yangians and classical Lie algebras, Russian Math. Surveys 51 205-282 (1996).
  • 21. Nazarov, M. and Tarasov, V.: Representations of Yangians with Gelfand-Zetlin bases, to appear in J. Reine Angew. Math. (q-alg/9502008).
  • 22. Opdam, E.: Harmonic analysis for certain representations of graded Hecke algebras, Acta Math., 175 75-121 (1995).
  • 23. Saito, Y., Takemura, K. and Uglov, D.: Toroidal actions on level-1 modules of $U_{q}(\hat{\mathfrak{sl}}_{N})$, Preprint RIMS-1133 (q-alg/9702024).
  • 24. Schoutens, K.: Yangian symmetry in conformal field theory, Phys. Lett. 331B 335-341 (1994).
  • 25. Sutherland, B.: J. Math. Phys. 12 246, 251 (1971); Phys. Rev. A4 2019 (1971); ibid. A5 1372 (1972).
  • 26. Takemura, K. and Uglov, D.: Level-0 action of $U_{q}(\hat{\mathfrak{sl}}_{N})$ on the q-deformed Fock spaces, Preprint RIMS-1096 (q-alg/9607031).
  • 27. Takemura K. and Uglov D.: The Orthogonal Eigenbasis and Norms of Eigenvectors in the Spin Calogero-Sutherland Model, J.Phys. A30 3685-3717 (1997).
  • 28. Uglov, D.: Semi-infinite wedges and the conformal limit of the fermionic spin Calogero-Sutherland model of spin $\frac{1}{2}$ , Nucl. Phys. B478 401-430 (1996).
  • 29. Uglov, D.: Yangian Gelfand-Zetlin bases, $\mathfrak{g}\mathfrak{l}_{N}$ -Jack polynomials and computation of dynamical correlation functions in the spin Calogero-Sutherland model, (hep-th/9702020).